Digital Signal Processing Reference
In-Depth Information
Note that the excitation or driving signal,
e
jωn
is a unity-magnitude complex exponential that
does not decay away, whereas the component due to
H(z)
, namely
C
S
a
1
, will decay away with increasing
n
if
|
a
|
<
1, i.e., if the system is stable.
Example 2.30.
Compute and plot the transient, steady state, and total responses to the complex expo-
nential
x
=
exp(jπ/
6
)
, and independently verify the total response, of the LTI system defined by the
following difference equation:
[
n
]
y
[
n
]=
x
[
n
]−
0
.
85
y
[
n
−
1
]
The following code generates Fig. 2.10:
w = pi/6; b0 = 1; a1 = - 0.85; Cs = b0*a1/(a1-exp(j*w));
Ce = b0/(1-a1*exp(-j*w)); n = 0:1:50; figure(11);
ys = Cs*a1.ˆn; yE = Ce*exp(j*w*n); yZ = ys+yE;
subplot(321); stem(real(ys)); subplot(322); stem(imag(ys));
subplot(323); stem(real(yE)); subplot(324); stem(imag(yE));
subplot(325); stem(real(yZ)); subplot(326); stem(imag(yZ));
1
1
0
0
−1
−1
0
10
20
30
40
50
0
10
20
30
40
50
(a) n
(b) n
1
1
0
0
−1
−1
0
10
20
30
40
50
0
10
20
30
40
50
(c) n
(d) n
1
1
0
0
−1
−1
0
10
20
30
40
50
0
10
20
30
40
50
(e) n
(f) n
Figure 2.10:
(a) Real part of transient response of IIR having a pole at -0.85 to a complex exponential
at frequency pi/6; (b) Imaginary part of same; (c) Real part of steady-state response; (d) Imaginary part
of steady-state response; (e) Real part of total response; (f ) Imaginary part of total response.
In Fig. 2.10, we see, in plots (a) and (b), the real and imaginary parts of the transient response.
Since the system pole that generates this response lies inside the unit circle, the system is stable, and this
system-generated response decays away. In plots (c) and (d), we see the steady-state response due to the
excitation function, a complex exponential. The total response is shown in plots (e) and (f ).
